When the dynamics are simple, a common approach to estimating \(F\)  is through a linear map (i.e. \(u(t+\tau )=Gu(t)\) ) using dynamic mode decomposition (DMD) [1]}. DMD works as a predictive tool for systems that decay to a fixed point or exhibit quasiperiodic dynamics with discrete frequencies. When applying DMD, or extended DMD (DMD with nonlinear observables), to nonlinear systems challenges arise in the form of closure of models and robustness to noise [2]}. Lusch et al. [3]} showed one way to extend these ideas for a few simple nonlinear systems is to use an autoencoder to learn a change of basis under which the dynamics are linear. Moreover, even with more complicated nonlinear systems there always exists a linear Koopman operator [4]}. The Koopman operator describes the evolution of arbitrary observables, but is infinite-dimensional. Due to these issues, we do not seek a strictly linear time evolution method. We instead consider methods that include nonlinearity.
